To compute and ``measure'' (with Monte Carlo) the critical exponents of
the model, in particular the critical exponent of the Helicity
Modulus .

Methods

Importance Sampling Monte Carlo (heat bath) ``with a twist'' to
get
at high precision for
.

Finite size scaling used to get critical exponents at (accepted value, ).

Helicity studied by freezing and twisting the (previously
periodic) boundary conditions in the (X,Y,1) plane.

Review of Theory

Landau potential for a continuous ferromagnetic model is:

(1)

where are the cartesian components of the coarse grain block
spins.

Define the block spin
in terms of its mean
value (the order parameter) plus a fluctuation:

(2)

Further decompose the fluctuation into a longitudinal and
transverse piece:

(3)

Derive the following general form for the free energy in terms of
the transverse coarse grained spin fluctuation gradient
:

(4)

(with phenomenological parameter , the ``spin wave stiffness''.

One can relate a state of uniform twist angle to the gradient of
the transverse spin fluctuation via
. Substituting and differentiating to find the free
energy density, one obtains the following two relations:

(5)

with

(6)

In Landau theory, approximately constant so
as from below. In detailed treatment one gets
corrections:

(7)

Finally, to use finite-size scaling theory (FSST) to extract the
critical exponent, we must substitute
, or

(8)

In the last expression,
.
term from clearly dominant ( is very small,
, for this model, while
and
). The helicity modulus should vanish sharply near
according to Landau theory.

But...

We cannot directly measure the free energy density . We can directly measure the enthalpy density . Following an identical
argument:

(9)

where
is the change in internal energy caused by
twisting the boundary conditions through the angle
with either helicity. From this obvious substitutions yield:

(10)

With a page or two of algebra we can show that:

(11)

with the critical exponent

(12)

This is what we wish to measure, in part to invert this equation
and deduce the values of and .

Note that as before, if we make the finite size scaling hypothesis we
will actually measure:

(13)

or

(14)

The enthalpy helicity should thus diverge at .

It is easy to show that:

(15)

(16)

(17)

where the second step uses ``hyperscaling'' (widely believed but by no
means proven for this model) to eliminate for . With
this we can compute and given and possibly
check hyperscaling.

Measuring with Monte Carlo

Calculations were performed on several generations of ``brahma''
(our beowulf compute cluster, also ganesh and rama).

Heat bath only (cluster method a bit difficult if boundary layers
are ``frozen'').

Equilibrate
lattice with periodic boundary
conditions.

``Freeze'' (x,y,1) layer of spins.

Rotate (x,y,1) spins through angle and store them in
(x,y,L+1) layer (replacing PBC's in z-direction with frozen twisted PBC's).

Re-equilibrate only the (x,y,2) to (x,y,L) spins with the heat
bath (with PBC's in the x and y directions).

Sample

Repeat (easiest to restore PBC's, re-equilibrate, repeat).

Sweep angles
, at .

Fit
where
.

Fit

Obtain , from hyperscaling.

Results

Figure 1:
E per spin as function of interlayer twist angle for (in progress). This is fit to obtain the helicity.

Figure 2: for and , evaluated at low
precision to get trend only.

Figure 3:
The helicities for various at . The nonlinear least
squares fit of this yields
.

Best result to date:

Conclusions

The only direct measurement of this quantity to date.

. This is quite large compared to most other Monte Carlo
results (which tend to yield
) but is not
inconsistent with the most recent renormalization predictions.

The hyperscaling relation itself then yields
.
This is a weakly singular quantity and is very difficult to
measure. This is a major motivation of this work.

For this particular talk, we emphasize that there are easily more
than 30 ``GFLOP-years'' of effort in this result (whatever you consider
a GFLOP to be). (32x400x3 = 38400) + (16x1300x2 = 41600) + (32x1600x1 =
51200) = 131.200 GHz-years, supercomputing indeed. Impossible without
the beowulf/cluster model.